Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userell ^1-spreading models in mixed Tsirelson space
100
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Suppose that (F_n)_{n=1}^{infty} is a sequence of regular families of finite subsets of N and (theta_n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T[(theta_{n}, F_n)_{n=1}^{infty}] is the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_0}, sup_n sup theta_n sum_{i=1}^{j}||E_{i}x||}, where the last supremum is taken over all finite subsets E_{1},...,E_{j} of N such that E_1 < >... <E_j and {min E_1,...,min E_j} in F_n. Necessary and sufficient conditions are obtained for the existence of higher order ell ^1-spreading models in every subspace generated by a subsequence of the unit vector basis of T[(theta_{n}, F_n)_{n=1}^{infty}.
rate research
Read More
We investigate the existence of higher order ell^1-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space X=T[(theta _n,S_n)_{n=1}^{infty}] (1)Every block subspace of $X$ contains an ell^1-S_{omega}-spreading model, (2)The Bourgain ell^1-index I_b(Y) = I(Y) > omega^{omega} for any block subspace Y of X, (3)lim_mlimsup_ntheta_{m+n}/theta_n > 0 and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions holds, then X is arbitrarily distortable.
Suppose that (F_n)_{n=0}^{infty} is a sequence of regular families of finite subsets of N such that F_0 contains all singletons, and (theta _n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). In this paper, we compute the Bourgain ell^1 - index of the mixed Tsirelson space T(F_0,(theta_n, F_n)_{n=1}^{infty}). As a consequence, it is shown that if eta is a countable ordinal not of the form omega^xi for some limit ordinal xi, then there is a Banach space whose ell^1-index is omega^eta . This answers a question of Judd and Odell.
If alpha and beta are countable ordinals such that beta eq 0, denote by tilde{T}_{alpha,beta} the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_{0}}, 1/2 sup sum_{i=1}^{j}||E_{i}x||}, where the supremum is taken over all finite subsets E_{1},...,E_{j} of $mathbb{N}$ such that $E_{1}<...<E_{j}$ and {min E_{1},...,min E_{j}} in S_beta. It is shown that the Bourgain $ell^{1}$-index of tilde{T}_{alpha,beta} is omega^{alpha+beta.omega}. In particular, if alpha =omega^{alpha_{1}}. m_{1}+...+omega^{alpha_{n}}. m_{n} in Cantor normal form and alpha_{n} is not a limit ordinal, then there exists a Banach space whose ell^{1}-index is omega^{alpha}.
The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space T[(S_n,theta_{n})_{n=1}^{infty}] and its modified version T_{M}[(S_{n},theta_{n})_{n=1}^{infty}]. It is shown that these spaces are not isomorphic for a large class of parameters (theta_{n}).
The unique maximal ideal in the Banach algebra $L(E)$, $E = (oplus ell^infty(n))_{ell^1}$, is identified. The proof relies on techniques developed by Laustsen, Loy and Read and a dichotomy result for operators mapping into $L^1$ due to Laustsen, Odell, Schlumprecht and Zs{a}k.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
